Signal structures for double side band-quadrature carrier modulation

ABSTRACT

Double side band-quadrature carrier modulation signal points are mapped on the complex plane are drawn from an alphabet consisting of at least 8 points, and are set up in concentric rings each rotated by 45* with respect to adjacent rings. Differential encoding is shown encoding the phase components of the transmitted signals.

United States Patent 1191 Forney, Jr. et al.

[ June 3, 1975 SIGNAL STRUCTURES FOR DOUBLE SIDE BAND-QUADRATURE CARRIERMODULATION [75] Inventors: George David Forney, Jr.; Robert G.

Gallager, both of Lexington, Mass.

[73] Assignee: Codex Corporation, Newton, Mass.

[22] Filed: Sept. 14, 1971 [21] Appl. No.: 180,289

[52] US. Cl 178/67; 325/30 [51] Int. Cl. H041 27/18 [58] Field of Search325/30, 49, 59, 60;

178/66 R, 67; 179/15 BC, 15 BM; 332/17 [56] References Cited UNITEDSTATES PATENTS 3,619,501 11/1971 Nussbaumer 178/67 l2 COMBINATIONALLOGIC A D/A 11/1971 Ragsdale 325/30 X 12/1972 Yanagidaira et a1. 178/66R Primary Examiner-Benedict V. Safourek [5 7] ABSTRACT Double sideband-quadrature carrier modulation signal points are mapped on thecomplex plane are drawn from an alphabet consisting of at least 8points, and are set up in concentric rings each rotated by 45 withrespect to adjacent rings. Differential encoding is shown encoding thephase components of the transmitted signals.

8 Claims, 15 Drawing Figures PULSES SEC IUULIL SIN w t SHEET 2 RIOR ARTPRIO% PRIOR ART PRIOR ART 4-(1) PSK e- PSK |6- FIG 20 FIG 2b FIG PRIORART PRIOR ART PRIOR ART I6-LEVEL 0 AM 4-,2-AMP| |TuDE 4-(1), 4-AMPLITUDE8 1 Z-AIVIPLITUDE FIG 2e FIG 2f FIG 9 n m I 3,8877% SHEET 3 Bl B3 B|B2B3BIETZB-IIS Bl B2B 3 BIB2B3 ET B2 l I l'l SIGNAL STRUCTURES FOR DOUBLESIDE BAND-QUADRATURE CARRIER MODULATION This invention relates to doubleside bandquadrature carrier (DSB-QC) modulation. DSB-QC modulationsubsumes a class of modulation techniques such as phase-shift-keying(PSK), quadrature amplitude modulation (QAM), and combined amplitude andphase modulation, such as have long been known in the art.

In high-speed data transmission across narrowbandwidth channels such asthe typical voice grade telephone channel, DSB-QC modulation has certaininherent advantages over single-sideband (SSB) and vestigial-sideband(VSB) techniques, such as are used in the majority of high-speed modemstoday. Against gaussian noise, it is inherently as efficient as SSB orVSB techniques in terms of the signal-to-noise ratios required tosupport a certain speed of transmission at a certain error rate in agiven bandwidth. In addition, a coherent local demodulation carrier canbe derived directly from the received data, without requiringtransmission of a carrier or pilot tone. Furthermore, DSB-QC systems canbe designed to have a much greater insensitivity to phase jitter on theline, or to phase error in the recovered carrier, than is possible with$88 or VSB signals.

For modest data rates, well-known modulation schemes such as four-phasemodulation provide good margins against both gaussian noise and phasejitter. At higher data rates, more bits of information must be sent persignalling interval, so multi-level signalling structures of greatercomplexity must be used. The standard schemes mentioned above begin todegrade rapidly against either gaussian noise or phase jitter when moresignal points are required. It is the principal purpose of the presentinvention to provide novel signal structures which continue to exhibitnear-optimum margins against both gaussian noise and phase jitter asadditional points are added. Further advantages of the invention aresimplicity of implementation and of detection, suppression of carrier,and 90symmetry, which allows use of differential phase techniques.

In general the invention features a double side bandquadrature carriermodulation system in which the signal points, as mapped on the complexplane, are drawn from an alphabet consisting of at least 8 points, andare set up in concentric rings each rotated by 45 with respect toadjacent rings. Preferred embodiments employ differential encoding ofthe phase components of the transmitted signals.

Other advantages and features of the invention will be apparent from thefollowing description of a preferred embodiment thereof, taken togetherwith the drawings, in which:

FIG. 1 is a block diagram of a DSB'QC modulation system;

FIGS. 2a-h show several prior art signal structures mapped on thecomplex plane;

FIGS. 3a, b show signal structures of the invention mapped on thecomplex plane;

FIGS. 4a, b are logic diagrams for implementation of the structures ofFIGS 3a, b;

FIG. 5 is a block diagram of a differential encoder; and

FIG. 6 is a block diagram of a receiver.

In DSB-QC modulation the transmitted spectrum X(w) is symmetric aboutsome center (carrier) frequency w In digital DSB-QC, data samples a'arrive at rates of l/T samples/second, and take on one of M valuesrepresented by a set of complex numbers 5,, l i M. Commonly M=2", and nbits can be transmitted per sample, or n/T per second. The transmittedsignal x(t) can be represented by where h(t) is the impulse response ofa low pass filter whose cutoff frequency is half the bandwidth of thechannel.

A circuit for realizing such a modulation scheme is shown in FIG. 1. A'stream of input bits arrives at a rate of n/T bits per second, and ispassed through an n-bit storage register 10. The n storage elements inthe register are inputs to a combinational logic circuit 12 which formsone of M=2" pairs of output words; this pair of words is a digitalrepresentation of the real and imaginary parts of the S appropriate tothe 11 bits of input. This pair of words controls a pair ofdigital'to-analog converters 14, 16, whose output voltages represent ReS; and Im 8,. Once each T seconds this pair of D/A outputs is gated toform a pair of narrow pulses of ampli tudes proprotional to Re 5, and ImS,-. Each of these pulse trains is then filtered in an identical linearfilter 18, 20 characterized by the impulse response h(t). F inally, thelower filter output is multiplied by sin(w t) (the quadrature carrier)and subtracted from the product of the upper filter output and cos(w r)(the in-phase carrier This is a baseband technique; there also existwell-known methods of operating directly on the carrier itself atpassband.

An aspect of the invention involves the realization that a signalstructure can be characterized by the sets of points [S l i M]associated with the modulation scheme, which we can map pictorially onthe complex plane. In PSK, for example, the M signal points aredescribed simply by a set of points evenly spaced around a circle. FIGS.2a, 2b, and 2c illustrate 4-, 8-, and 16- phase modulation according tothis method of representation. In QAM, Re S,- and Im S may each take onindependently one of m levels, typically equallyspaced, so that M=mFIGS. 2d and 2e illustrate 4-level and l6-level QAM; it will be notedthat 4-level QAM is effectively identical to 4-phase PSK in thisrepresentation, although their implementations may be quite different.Finally, in combined amplitude and phase modulation, the amplitude and!phase variables are independently varied, to give for example the4-phase and 2- or 4-amplitude structures of FIGS. 2f and 2g, or the8-phase, Z-amplitude structure of FIG. 2h.

This method of representation permits examination of the effect ofdisturbances on the modulated waveform x(t). We first consider an idealcase, illustrated in FIG. 6. x( t) enters the receiver and isdemodulated by the two locally-generated carriers cos w t and sin w t.The double-frequency terms at 2 w are removed by low-pass filters 30, 32to recover the low pass in-phase and quadrature waveforms Now supposethat 11(1) is a perfect Nyquist waveform, i.e., for some time, T,l1(r)=l, but lz(r-kT)=O for integers k O or k 0. Then if we sample thetwo channels every T seconds at the correct times 'rl-kT, there will beno intersymbol interference, and we simply recover the pair of voltagesRe z Re a' and Im 1,, Im d which tell us which bits were sent.

In a real situation, h(t) will not be a perfect Nyquist waveform, andthe channel will introduce additional linear distortion which will leadto intersymbol interference. (At high data rates, it is usuallynecessary to in- (r). The effect of such a phase error is to rotate thereceived vector in the complex plane by the phase angle 6 0 (r+kT), sothat the received complex value is where z is the value which would havebeen received had there been no phase error. It is therefore especiallyimportant that signal points be well-separated in phase.

Table 1 below gives required signal-to-noise ratios and minimum phaseseparations of points of the same amplitude for the signal structures ofFIGS. 211-11. (The minimum phase separation criterion above is anoversimplified. but still qualitatively indicative, measure of phasejitter immunity, since errors will actually be caused by the combinedeffects of noise and phase jitter.)

Table I 2a 2b 2c 2d 2e 2f 2g 2h Required Signalto-Noise Ratio (dB) 3 8.3l4.l 3 II) 8.4 l3.9 11.5

Phase Separation 90 45 215 90 37 90 90 45 then e is equally likely to bea vector of any phase. Against such disturbances, therefore, we realizeit to be desirable to maximize the Euclidian distance between signalpoints, subject to a constraint in the total signal energy E, defined asWe define the required signal-to-noise margin S as 10 log E dB, where Eis calculated for the signal points S,- scaled so that the minimumEuclidean distance between any two points is 2 (so that an error canoccur only if [2,, 2 1).

Another disturbance of importance on telephone lines is phase jitter. Ifa transmitted waveform .\'(t) is subject to phase jitter, the result is(to first order when the phase jitter is slow and channel filteringunimportant) where 6(t) is a random phase process. Typically ontelephone lines 6(t) contains frequencies up to 180 Hz, and may haveamplitude up to 30 peak-to-peak or more. To some extent the phase jittercan be tracked at the receiver to give the locally-generated carrierscos(w t 0(t)) and sin(w t 0'(t)), but there will always remain someresidual phase error 0,.(t) 0(t) Experience has shown that on telephonelines a minimum phase separation of 45 may be insufficient to guaranteelow error rates when phase jitter is severe. For M=8 or 16, this meansthat only the 4-phase, 2- or 4-amplitude structures of FIGS. 2f and 2gcan be used. But these structures are rather inefficient in their use ofpower, as is shown by their values of required signalto-noise margin inTable l.

The signal structures of the present invention retain the full phaseseparations of the 4-phase structures, as well as their four-phasesymmetry, while substantially reducing the required signal-to-noisemargin over the structures of FIGS. 2f and 2g. FIG. 3a illustrates astructure according to the invention for the case M=8, and FIG. 3b, astructure for M=l6. In the former case the points are at (l-l-j)j"' and3j" for lr=0,1,2,3; in the latter case they are at these eight pointsplus the points 3(l+j)j" and 5j", l\=0, l, 2, 3. FIG. 3a resembles the4-phase, 2-amplitude structure of FIG. 2f, except that the two ringshave been rotated 45 with respect to one another, which allows the outerradius to be decreased without loss of signal-to-noise margin. (Actuallythe outer ring could be pulled in slightly more, but use ofinteger-valued coordinates simplifies implementation.) Similarly, FIG.3b resembles FIG. 2b, except that the second ring is rotated 45withrespect to the first, the third 45 with respect to the second, and thefourth 45 with respect to the third, allowing decreases in the radii ofall outer rings without loss of signaI-to-noise margin.

Table II below gives required signal-to-noise ratios and minimum phaseseparations over the structure of FIGS. 3a and 3b. The savings overFIGS. 2f and 2g are 1 dB and 2.6 dB, respectively. In fact FIG. 3b isonly 1.3 dB worse than the optimal FIG. 2c for M=16, but has greatlyenhanced protection against phase errors.

In general, the class of structures according to the invention may bedescribed as follows. Interest is confined to M-point structures for M 28, since the simple 4-phase structure of FIG. 2a is entirelysatisfactory for M=4. M is assumed to be a multiple of 4, as it will beif it is a power of 2. Then, m=M/4 rings of radii r r r are set up, withfour points on each ring, and with each succeeding ring rotated 45withrespect to the previous one. The set {8,} may be described gener ally bythe complex numbers ar u j where l s i m, s k s 3, Mi 1 forieven andl+j/\ for i odd, and a is an arbitrary complex constant. In some of theouter rings it may be aceptable to use 8-phase structures; thispossibility is accounted for by the requirement r r s r r s r,,,; thusonly the innermost ring necessarily contains four points.

Implementation of the invention is straight-forward. The circuit of FIG.1 can be used with appropriate combinational logic to generate theintegers 0, +1, +3, or +5 in ordinary twos-complement form, which canthen drive standard 3- or 4-bit D/A converters. FIG. 4a givesappropriate logic for the signal structure of FIG. 3a, where (B1, B2,B3) are the three input bits, (XS, X1, X2) and (Y5, Y1, Y2) aretwoscomplement representations of the real and imaginary parts of thesignal points, and the correspondence is according to the three-bitnumbers associated with each signal point on the diagram of FIG. 3a. (Inthis correspondence B1 is in effect an amplitude variable denoting inneror outer ring, whereas B2 and B3 select one of the four phases.)Similarly FIG. 4b gives logic for FIG. 3b, where (B1, B2, B3, B4) arethe four input bits and (XS, X1, X2, X3) and (Y5, Y1, Y2, Y3) are thecoordinates of the signal points in twos-complement form, codedaccording to the diagram of FIG. 3b (where B1 and B2 select one of thefour rings, and B3 and B4 select the phase on the ring).

Because of the four-phase symmetry of these structures, the carrier issuppressed-i.e., there is no carrier power at the frequency wNonetheless there are a number of techniques by which a carrier may bederived by the receiver from the received data signal. Such techniquesgenerally cannot distinguish between the correct phase of the receivedcarrier and the cor rect phase plus multiples of 90, due again to the 90symmetry of the signal structure, and so may set up in any of fourphases; there is said to be 90 phase ambiguity in the recovered carrier.It is advantageous under these conditions to differentially encode thephase of the transmitted signal, by selecting the phase of the signaltransmitted at time t on the basis of the bits for time 1 and the phasetransmitted at time tl. For example, in the eight-point structure ofFIG. 3a, the two bits B2 and B3 select the phase of the transmittedsignal according to where d(0)=(1+j) and d( l 3, while 0(0, 0) =0, 0(0,1)=1r/2,6(1,1)=1r, and 0( l, 0) 37r/2, and B1,, B2,, and B3,, representthe values of the three input bits at time k. If instead the phase isdifferentially encoded then the phase 9,, at the time k is made equal tothe phase 6 at time kl plus 0(B2 B3 i.e.,

Then at the receiver the phase 0(B2,,., 83 is detected as the differencebetween the estimates 0,,- and 6,,. and is unaffected by constant phaserotations. The same differential phase technique can be used with thephase bits B3 and B4 of FIG. 3b, or indeed with any of the signalstructures of the invention.

FIG. 5 illustrates the implementation of differential encoding. Thephase bits B2 and B3 are Gray-coded into a 2-bit integer which is addedto the stored Z-bit integer (0l,,. 02,,. without carryi.e., modulo 4.The result is an integer (01 92 representing the current phase, which isstored in a 2-bit memory after each sample by a clock pulse (not shown),to become the integer (0l 62 for the next sample. The integer is alsoGray-decoded to form a (B2,B3) which can be used instead of (B2, B3) asthe input to the combinational logic of FIG. 4a. [Note that when (0l02,,. (O, 0), (B2', B3) (B2, 83).]

Other embodiments are within the following claims:

We claim:

1. A double side band-quadrature carrier modulation system comprisinginput means for receiving a sequence of symbols (1,,-

at a rate l/T per second. coding means connected to said input means forproviding from said symbbols a sequence of complex valued signal pointsd drawn from an alphabet comprising M points arranged in a multiplicityof concentric rings in the complex plane including an innermost ringhaving four equally spaced points and a plurality of additional ringseach having four equally spaced points, each said ring being rotated by45 with respect to adjacent said rings, and

modulating means connected to said coding means for providing from saidsignal points a signal in the form where h(tkT) represents an impulseresponse, w represents a carrier frequency, t represents time, j equalsV ll, and k is the index of d and a 2. The system of claim I. whereinsaid coding means includes means for effectively providing said signalpoints arranged in at least four concentric rings inthe complex plane.

3. The system of claim 2 wherein said coding means includes means forcausing the innermost four said rings to have radii in the ratio V2:3:3V25.

4. The system of claim 1 wherein said coding means includes means forcausing the phase component of each (1;, to depend upon a and upon thephase component of d 5. The system of claim I wherein said coding meansincludes means for causing each said (1,,- to have integer valuedcoordinates in the complex plane.

6. A double side band-quadrature carrier modulation system comprisinginput means for receiving a sequence of symbols a at a rate l/T persecond,

coding means connected to said input means for providing from saidsymbols a sequence of complex valued signal points d drawn from analphabet 7 comprising M points arranged in a multiplicity of concentricrings in the complex plane including an innermost ring having fourequally spaced points and a plurality of additional rings having fourequally spaced points, each said ring being rotated by 45 with respectto adjacent said rings, and filtering means connected to said codingmeans for providing from said signal points the real and imaginary partsof a complex valued baseband signal in the form and modulating meanseffectively connected to said filtering means for providing from saidbaseband signal a passband signal in the form where h(tkT) represents animpulse response, w represents a carrier frequency, t represents time, jequals V -1, and k is the index of d and a 7. A double sideband-quadrature carrier modulation method comprising receiving asequence of symbols a at a rate l/T per second providing from saidsymbols a sequence of complex valued signal points d drawn from analphabet comprising M points arranged in a multiplicity of concentricrings in the complex plane including an innermost ring having fourequally spaced points and a plurality of additional rings each havingfour equally spaced points, each said ring being rotated by 45 withrespect to adjacent said rings, and providing from said signal points asignal in the form where h(tkT) represents an impulse response, wrepresents a carrier frequency, t represents time, j equals 1, and k isthe index of d and a 8. A double side band-quadrature carrier modulationmethod comprising receiving a sequence of symbols a at a rate l/T persecond, providing from said symbols a sequence of complex valued signalpoints d drawn from an alphabet comprising M points arranged in amultiplicity of concentric rings in the complex plane including aninnermost ring having four equally spaced points and a plurality ofadditional rings having four equally spaced points, each said ring beingrotated by 45 with respect to adjacent said rings, and providing fromsaid signal points the real and imaginary parts of a complex valuedbaseband signal in the form and providing from said baseband signal apassband signal in the form where h(t-kT) represents an impulseresponse, w represents a carrier frequency, t represents time, j equalsl, and k is the index of d and a Page '1 of 9 UNITED STATES PATENT ANDTRADEMARK OFFICE CERTIFICATE OF CORRECTION PATENT NO. 1 3,887,768

DATED 1 June 3, 1975 I Geor e David Forne Jr INVENTOR s g Y Robert G.Gallager It is certified that error appears in the above-identifiedpatent and that said Letters Patent are hereby corrected as shown below:

--3,l23,670 3/1964 Kaenel 178/66'R-- should be added to the list of"References Cited" Abstract, line 2 "are", first occurrence, should beas- Col. 2, line 32, "proprotional" should be "proportional- Col 3, line40, "Euclidian" should be "Euclidean-- 7 C01. 4, line 55, "over" shouldbe --for- Col. 5, line 13, "dceptable" should be "ecceptable" Col. 5,line 50, ilisert -abefore "90" Col. 6, claim 3, line 54, "m r TS" shouldbe .--/:3:3/7: 5--

Col. 7, claim 6, line 16, "effectively" should be deleted C01. 8, claim7, line 7, "l" should be "FT-- Q, Col. 8, claim 8, line 37 "-l" shouldbe "m" read as shown below.

Page 2 of 9 UNITED STATES PATENT AND TRADEMARK OFFICE CERTIFICATE OFCORRECTION PATENT NO. 3,887,768 DATED June 3, 1975 |NVENTOR(S) GeorgeDavid Forney, Jr.

Robert G. Gallager I I It is certified that error appears in theabove-rdentlfred patent and that sald Letters Patent are herebycorrected as shown below:

The symbol "d" should have been printed in italics with the underliningomitted at: Col. 2, lines 10, 12 (two occurrences); Col. 6, line 42;C01. 7, lines 12, 20;

C01. 8, lines 1, 25, 32

The lower case letters used to designate FIGURES of the drawings shouldnot have been italicized at: Col. 1, lines 59, 61, 63, 64; C01. 2, lines47, 51, 58, 59; C01. 4, lines 11, 29, 37, 39, 42, 43, 49, 56, 57, 58;C01. 5, lines 4, 22, 24, 30, 33, 37, 56; C01. 6, lines 8, 20

The following mathematical symbols and expressions should Theunderlining of a symbol indicates that said symbol should have beenprinted in italics with the underlining omitted. All symbols notunderlined should have been printed without italics.

Col. 2, line 3, --w --d line 5, s

line 6, --l i M-- -M=2 line 8, --x(t)- line 17, --h(t) line 2 2, --n/T--line 25, --M=2 line 27, -S

Page 5 of 9 UNITED STATES PATENT AND TRADEMARK OFFICE CERTIFICATE OFCORRECTION Q PATENT NO. 3,887,768

DATED June 3, 1975 |NVENTOR(5) George David Forney, Jr.

' Robert G. Gallager ltis certified that error appears in theabove-identified patent and that said Letters Patent Q are herebycorrected as shown below:

Col. 2, line 29, --Re-- line 30, -ImS O i i line 32, --Re --ImS i i Line34, --h(t) line 35, -sin(w t)- line 37, --cos(w t)-- line 43, [Q lsisM]1 line 49, -Re --Im 9 i 1 line 50, -m-

line 51, --M=m line 62, -x(t)-- line 63, -x(t)- line 64, -cos (w t)sin(w t) line 65, -Z w t C Col. 3, line 1, -Z2Re 1 h(t-kT) Page of 9UNITED STATES PATENT AND TRADEMARK QFFICE CERTIFICATE OF CORRECTIONPATENT NO. 3,887,768

DATED June 3, 1975 |NVENTOR(S) George David Forney, Jr.

' Robert G. Gallager It is certified that error appears in theabove-identified patent and that said Letters Patent are herebycorrected as shown below:

C01. 3, line 4, "21mg h(tkT)- k line 6, -h(t)-,- line 7, --T, h(T)=1--,--h(T-kT)=O-- line a, --1 0 or 1 0-- line 9, --T kT-- line 11, "Reg =Rel and Im5 =Im d k k k k line 13, --h(t) line 33, --Re z k line 34, --Imz k line 35, "g

k line 37, -e =z d k k k a line 38, "g

1 M line 45, --E 2 is l M i=1 1 line 49,

Page 5 of 9 UNITED STATES PATENT AND TRADEMARK OFFICE CERTIFICATE OFCORRECTION PATENT N0. 1 3,887 ,768 DATED June 3, 1975 |NVENTOR(S) GeorgeDavid Forney, Jr.

' Robert G. Galla er It IS certified that error appears in the aave-identified patent and that said Letters Patent are hereby correctedas shown below:

Col. 3, line 52, lg ,1--

k line 54, --x(t)-- jt r n line 59, --x' (t) ReZ c 1 h(t-kT) e line 61,"9 (t) line 62, -6(t)-- line 66, -cos (w t 9 (t)) and sin(w t 6 (t)),--

Col. 3, line 67 to C01. 4, line 1, --9 (t) 6'(t) 6(t)-- Q C01. 4, line3, 6 k 6 (T+kT) (3' line 5, z e ek z line 6, -z "k line 38, -M=8- line39, --M=16- line 40, 1+ and 3 3 for 1 =0, 1, z, 3;-- tline 42, --3(1+j)jand- Sj 1 =0, 1 2, 3.--

line 58, -FIG. 2e-, --M=l6- Page 6 of E UNITED STATES PATENT ANDTRADEMARK OFFICE CERTIFICATE OF CORRECTION 3,887,768 June 3, 1975 GeorgeDavid Forney, Jr. Robert G. Gallager It IS certified that error appearsin the above-identified patent and that said Letters Patent are herebycorrected as shown below:

PATENT NO.

DATED INVIENTOR(S) C01. 5, line 5, -M=4-- line' 6, --m=M/4-- line 6 to 7"r r ,r

line 9, S I" line 11, where 1 i m, 0 1 3, u =1 for iline 12, --i-- line15, --r .s'r

line

line 42, w

line 54, --t-- line 55, --t-- -t1-- line 60, -.-g

line 62, 1(0) lines 62 to 63,

iand e 1,o 31r/2, and 131 132 Page 7 of 9 UNITED STATES PATENT ANDTRADEMARK OFFICE CERTIFICATE OF CORRECTION PATENT NO. 3,887,768

DATED June 3, 1975 |NVENTOR(S) George David Forney, Jr.

' Robert G. Gallager It rs certified that error appears in theabove-identified patent and that said Letters Patent Q are herebycorrected as shown below:.

Col. 5, line 64, -B3

line 65, -k-- 0 line 66, "6 --k line 67, --9 at time k-l plus 6(B2 ,B3

' Col. 6, line 1, 6 9 6(B2 ,B3

3 line 2, :'l @(BL )e line 4, 6(BZ ,B3

line 5 "6 line 6, --6 line 13, --(61 ,6Z

line 14, -(6l ,62

line 17, --(61 ,62

lines 20 to 21, -(61 ,62 (0,0) (B2' ,B3') Q k l k 1 8 C01. 6, claim 1,line 26, --a

Page 8 of 9 UNITED STATES PATENT AND TRADEMARK OFFICE CERTIFICATE OFCORRECTION PATENTNO.: 3,887,768 DATED June 3, 1975 |NvE (5) George DavidForney, Jr.

' Robert G. G al1-a er I It Is certrfred that error appears In the aove-identrfied patent and that sald Letters Patent are hereby correctedas shown below: 5

C01. 6, claim 1, line 27, --1/T- line 30, "Q

line 45, -h(t-kT)--, 'w

line 46, -t, --j

line 47, --1 Q -a C01. 6, claim 4, line 57 "Q -a line 58 1 Col. 6, claim5, line 60, Q

C01. 6, claim 6, line 64, --a

line 65, -1/T-- line 68, "Q

C01. 7, claim 6, line 23, -h(t-kT)--, --WC-- line 24,

line 25, --k-- "g Page 9 of 9 UNITED STATES PATENT AND TRADEMARK OFFICECERTIFICATE OF CORRECTION PATENT NO. 3,887,768 DATED June 3-, 1975 v 0(5) George David Forney, Jr.

' Robert G. Gallager It is certified that error appears in theabove-identified patent and that said Letters Patent are herebycorrected as shown below:

Col. 7, claim 7, line 28, -a

line 31, "Q

Col. 8, claim 7, line 5, --h(t-kT)-, --W

line 6, -t, j-

line 7, --k--, --d

line 36, -t--, --j

line 37, --k-, d --a Signed and Scaled this.

Twelfth D3) Of October 1976 A ttes t:

RUTH C. MASON Arresting Officer C. MARSHALL DANN Commissioner oj'Parentsand Trademarks

1. A double side band-quadrature carrier modulation system comprisinginput means for receiving a sequence of symbols ak at a rate 1/T persecond. coding means connected to said input means for providing fromsaid symbbols a sequence of complex valued signal points dk drawn froMan alphabet comprising M points arranged in a multiplicity of concentricrings in the complex plane including an innermost ring having fourequally spaced points and a plurality of additional rings each havingfour equally spaced points, each said ring being rotated by 45* withrespect to adjacent said rings, and modulating means connected to saidcoding means for providing from said signal points a signal in theform
 1. A double side band-quadrature carrier modulation systemcomprising input means for receiving a sequence of symbols ak at a rate1/T per second. coding means connected to said input means for providingfrom said symbbols a sequence of complex valued signal points dk drawnfroM an alphabet comprising M points arranged in a multiplicity ofconcentric rings in the complex plane including an innermost ring havingfour equally spaced points and a plurality of additional rings eachhaving four equally spaced points, each said ring being rotated by 45*with respect to adjacent said rings, and modulating means connected tosaid coding means for providing from said signal points a signal in theform
 2. The system of claim 1 wherein said coding means includes meansfor effectively providing said signal points arranged in at least fourconcentric rings in the complex plane.
 3. The system of claim 2 whereinsaid coding means includes means for causing the innermost four saidrings to have radii in the ratio Square Root 2:3:3 Square Root 2:5. 4.The system of claim 1 wherein said coding means includes means forcausing the phase component of each dk to depend upon ak and upon thephase component of d(k 1).
 5. The system of claim 1 wherein said codingmeans includes means for causing each said dk to have integer valuedcoordinates in the complex plane.
 6. A double side band-quadraturecarrier modulation system comprising input means for receiving asequence of symbols ak at a rate 1/T per second, coding means connectedto said input means for providing from said symbols a sequence ofcomplex valued signal points dk drawn from an alphabet comprising Mpoints arranged in a multiplicity of concentric rings in the complexplane including an innermost ring having four equally spaced points anda plurality of additional rings having four equally spaced points, eachsaid ring being rotated by 45* with respect to adjacent said rings, andfiltering means connected to said coding means for providing from saidsignal points the real and imaginary parts of a complex valued basebandsignal in the form
 7. A double side band-quadrature carrier modulationmethod comprising receiving a sequence of symbols ak at a rate 1/T persecond providing from said symbols a sequence of complex valued signalpoints dk drawn from an alphabet comprising M points arranged in amultiplicity of concentric rings in the complex plane including aninnermost ring having four equally spaced points and a plurality ofadditional rings each having four equally spaced points, each said ringbeing rotated by 45* with respect to adjacent said rings, and providingfrom said signal points a signal in the form